Message 3

From: "Andrey Kulsha"

Andrey_601@...
Date: Tue May 16, 2006 0:08pm(PDT)

Subject: Re: A surprising algebraic factorization

> 4 x^4 + y^4 = ( 2 x^2 - 2 x y + y^2) ( 2 x^2 + 2 x y + y^2)

http://xyyxf.at.tut.by/aurifeuillean.pdf
Which contains many identities such as

A^2 + B^2 = {1, 1}^2 2AB

Thanks.

I figure out the notation thusly.

(A + B)^2 = {1, 1}^2

The second identity in the

http://xyyxf.at.tut.by/aurifeuillean.pdf
Is

(A^3 + B^3)/((A+B) = {1, 1}^2 - 3 A B

A^3 + B^3 = [ {1, 1}^2 - 3 A B ] (A + B) = {1,1}^2 - 3 A B (A + B)

(A + B)^3 = {1,1}^2 (A + B)

(A + B)^2 = {1,1}^2

All of the identities in that page are based on the difference of odd powers

The identity

> 4 x^4 + y^4 = ( 2 x^2 - 2 x y + y^2) ( 2 x^2 + 2 x y + y^2)

is surprising in that the difference of squares is the same as the sum of

4th powers.